Journal of Physics Special Topics
A4 6 The Penny Drops
J.C. Coxon, J.F. Barker and T.M. Conlon
Department of Physics and Astronomy, University of Leicester. Leicester, LE1 7RH.
Feb 16, 2011.
Abstract
A British 1p coin is dropped from a great height. Its terminal velocity is calculated assuming that it falls
face-down, alongside the force (and subsequently pressure) it would exert upon a surface on landing. The
value is compared to the tensile strength of normal human skin in order to examine the effects it might
have if it landed on a person, and is found not to be high enough to break the skin.
Introduction
A common piece of folklore imagines someone standing atop a tall skyscraper (commonly, the Empire State
Building) and dropping a penny. This penny then collides with a pedestrian below, killing them instantly.
This paper aims to consider whether this really is the
outcome of such a scenario by considering the properties of a penny and the physics of its fall to calculate
the force (and the pressure) inflicted on human skin
when the penny lands on its victim.
the penny vt from Eq. (3). Starting with the equations
of motion, bearing in mind that the initial velocity u =
vt and the final velocity v = 0 ms−1 , and also assuming
a constant deceleration on impact,
v = u + at
(4)
v = 0 ⇒ u = vt = −at
(5)
vt
(6)
⇒ −a =
Terminal Velocity
t
When an object falls through the air, it has two forces
where t is the time it would take the penny to deacting on it. The first is the force due to gravity,
celerate upon hitting the skin. Using the relationship
Fg = mg,
(1) between deceleration a and impact force Fi , we can derive an equation to find this force and an equation to
where m is the mass and g is the acceleration due to
find the pressure exerted on impact P (where Ai is the
gravity (on Earth, g = 9.81 ms−2 ).
area of the penny that impacts upon the skin):
The second force that has an effect on the penny is
ma
mvt
the drag due to the fluid
Fi = ma ⇒ Pi =
=
(7)
A
tAi
i
2
Cd ρa Av
Fd =
,
(2)
Discussion
2
The above equations assume knowledge of several conwhere Cd is the coefficient of drag; ρa is the density of
stants, which are as follows. The mass m, diameter
air; A is the area of a penny and v is the velocity of
2r and thickness s of a British 1p coin are given by
the object moving. In this paper, it will be assumed
[1] as m = 3.56 × 10−3 kg, 2r = 2.03 × 10−2 m and
that the penny will fall face-down and as such the area
s = 1.52 × 10−3 m respectively. The drag coefficient
of the penny A = πr2 (the case in which the penny
of a thin disc Cd = 1.1 [2] and the density of air
falls with its edge facing the ground is left for further
ρa = 1.29 kg m−3 [3].
investigation).
Using these values for the constants, a terminal veEq. (2) shows that the force due to drag on the
locity is derived from Eq. (3) as vt = 12.3 ms−1 . This
penny will increase as it accelerates until the gravity
value can be put into Eq. (6) and then Eq. (7) to caland drag forces are equal, at which point, the termiculate values for the impact force Fi and the pressure
nal velocity vt has been reached. An equation for this
exerted Pi .
velocity is derived by equating Eq. (1) and Eq. (2):
However, the time taken for the penny to decelerate
r
2 2
t
must
be assumed. Since it is assumed that the imCd ρa πr vt
2mg
⇒ vt =
.
(3) pact would be visible on a high-speed camera (which
mg =
2
2
Cd ρa πr
typically record at 1000 fps [4]) and since it is desirable
Landing
to minimise the time taken to decelerate to provide the
An expression for the deceleration a of the penny on highest possible force and pressure exerted on the skin,
landing can be derived using the terminal velocity of a value of t = 10 ms is assumed. The impact area of
p-1
A4 6 The Penny Drops, Feb 16, 2011.
the penny with the skin Ai is also assumed (in this
paper, Ai = s2 = 2.31 × 10−6 m2 is used).
In this scenario, it is found that Fi = 4.37 N and
Pi = 1.89 MPa.
Conclusion
It is known that the average tensile strength of human
skin P = 20.89 ± 4.11 MPa [5]. The calculated value
for the pressure exerted on the person upon landing,
Pi = 1.89 MPa, is much lower than this value. The
penny’s impact with its victim would not even break
the skin.
REFERENCES
[1] royalmint.com/Corporate/facts/coins/1pCoin.aspx
[2] engineeringtoolbox.com/drag-coefficient-d 627.html
[3] P.A. Tipler and G. Mosca, Physics for Scientists and
Engineers: Fifth Edition (W.H. Freeman and Company, 2003), Appendix B, p AP-3.
[4] http://amazon.com/Casio-Exilim-EX-FH20-DigitalOptical/dp/B001G0N5H8
[5] J. Jussila, A. Leppäniemi, M. Paronen and E. Kulomäki, Forensic Science International 150, p63 (2005).
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